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首页> 外文期刊>International journal of computer mathematics >Construction of symmetric or anti-symmetric B-spline wavelets and their dual wavelets
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Construction of symmetric or anti-symmetric B-spline wavelets and their dual wavelets

机译：对称或反对称B样条小波及其对偶小波的构造

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摘要

Suppose N_m(x) is the B-spline function of order m. An explicit construction algorithm for the wavelet with symmetry φ_(m,n)(x):=2~(-n+1)∑~n_(e=0)(-1)~e n!/e!(n-e)! N_m(2x-e) associated with N_m (x) is presented, where n is an arbitrary positive integer and 4 does not divide (m + n). By appropriately selecting n, we can obtain the B-spline wavelet with short support or arbitrarily high vanishing moments. When 4 does not divide m+1, we prove that φ_m,1(x) corresponding to N_m(x) has the shortest support among the wavelets whose scaling functions have an approximation of order m. Moreover, the dual scaling function N_m(x) and the dual wavelet φ~~_(m,n)(x) are also constructed explicitly. Thereby, N~~_m(x) and φ~~_(m,n)(x) are symmetric or anti-symmetric. Furthermore, we study the regularity of N~~_m(x). Particularly, we find that as n increases, the order of vanishing moment of φ_(m,n)(x) as well as the regularity of N~~_m(x) also increases. Two examples are given to illustrate our results.
机译：假设N_m（x）是阶m的B样条函数。对称性φ_（m，n）（x）：= 2〜（-n + 1）∑〜n_（e = 0）（-1）〜e n！/ e！（n-e）！给出了与N_m（x）关联的N_m（2x-e），其中n是任意正整数，而4不除（m + n）。通过适当地选择n，我们可以获得具有短支撑或任意高消失力矩的B样条小波。当4不除m + 1时，我们证明与N_m（x）对应的φ_m，1（x）在缩放函数近似为m阶的小波中支持最短。此外，对偶缩放函数N_m（x）和对偶小波φ〜__（m，n）（x）也被明确构造。因此，N〜__m（x）和φ〜__（m，n）（x）是对称的或反对称的。此外，我们研究了N〜__m（x）的规律性。特别地，我们发现随着n的增加，φ_（m，n）（x）消失力矩的阶数以及N〜_m（x）的规律性也增加。给出两个例子来说明我们的结果。

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